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  "body": "\n## Statement\n\nLet $(X, \\Sigma, \\lambda)$ be a measure space with $\\lambda$ a [[finite measure|63Z1Q4FS23]].\nLet $(\\mu_n)_{n \\ge 1}$ be a sequence of [[complex measures|NY6TYHRGMS]] on $\\Sigma$, each [[absolutely continuous|P0P21KDV6E]] with respect to $\\lambda$.\nAssume that for every $E \\in \\Sigma$ the limit $\\mu(E) = \\lim_{n \\to \\infty} \\mu_n(E)$ exists in $\\C$.\nThen:\n\n1. The $\\mu_n$ are uniformly absolutely continuous with respect to $\\lambda$: for every $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $\\lambda(E) \u003c \\delta$ implies $\\abs{\\mu_n(E)} \u003c \\varepsilon$ for all $n$.\n2. The set function $\\mu$ is a complex measure on $\\Sigma$, and it is absolutely continuous with respect to $\\lambda$.\n\n## Remarks\n\nA standard proof uses the Baire category theorem.\nThe measurable sets modulo $\\lambda$-null sets form a complete metric space with distance $d(E, F) = \\lambda(E \\,\\triangle\\, F)$.\nEach $\\mu_n$ is continuous on this space, and a Baire argument on the sets where $\\abs{\\mu_n(E) - \\mu_m(E)} \\le \\varepsilon$ for all $m, n \\ge N$ gives the uniform bound.\n\nExample: let $f_n \\in L^1(\\lambda)$ with $\\int_E f_n \\, d\\lambda$ convergent for every $E$.\nThe theorem applied to $\\mu_n = f_n \\lambda$ shows that $(f_n)$ is uniformly integrable, which is a key step in the Dunford–Pettis theorem.\n\nThe hypothesis asks for convergence on every set of $\\Sigma$.\nConvergence on a generating family of sets, such as intervals, is not enough in general.\n\n## Explanations\n\n### L1 human\n\nIf a sequence of measures converges on every single set, then the limit is again a measure.\nMoreover, the measures cannot concentrate mass on smaller and smaller sets at different speeds: they are controlled uniformly by the reference measure.\n\n### L2 human\n\nThe theorem is the measure version of the uniform boundedness principle: pointwise convergence on a complete space forces a uniform estimate.\nIt is used to characterise weakly convergent and weakly compact sets in $L^1$.\nThe Nikodym convergence theorem is a version that needs no reference measure $\\lambda$.\n",
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